Researchers published a semi-analytical study in Discover Nano examining the viscous flow and heat transfer characteristics of a tri-hybrid nanofluid over a spinning disk. The investigation focuses on the GO-Cu-TiO2 nanofluid suspended in engine oil, analyzing how thermal radiation, thermal stratification, and porous media effects influence the fluid dynamics. The study utilizes the Darcy-Forchheimer model to represent flow through a porous substrate, incorporating second-law thermodynamic analysis to evaluate entropy generation and the Bejan number. This work provides a comprehensive mathematical framework for understanding complex thermal management systems involving multiple nanoparticle types.
The research team, led by S. S. Zafar and colleagues, employed the Optimal Homotopy Analysis Method (OHAM) to solve the governing ordinary differential equations. By applying similarity transformations, the researchers reduced the partial differential equations describing the flow to a set of solvable ordinary equations. The study specifically investigates the convergence and residual error of the OHAM solution, ensuring the mathematical validity of the results. The authors compared their computed values with previously published data, reporting excellent agreement that validates the accuracy of the proposed numerical approach.
The physical model assumes a steady, incompressible, and axisymmetric flow of the tri-hybrid nanofluid past a porous spinning disk. The disk rotates around the z-axis at a constant angular velocity, while the flow is described in cylindrical coordinates. The base fluid is engine oil, into which graphene oxide, copper, and titanium dioxide nanoparticles are evenly distributed. The model accounts for thermal radiation and thermal stratification, where the ambient temperature varies with radial distance, providing a more realistic depiction of real-world engineering systems than constant-temperature assumptions.
The Darcy-Forchheimer model is used to simulate fluid motion across porous materials at medium to high speeds, where viscous impacts are significant. Unlike the standard Darcy law, which applies to slow creeping flows, the Forchheimer term incorporates a quadratic inertia barrier that accounts for fluid inertia. This addition allows for a more accurate representation of velocity transport through porous objects. The model is relevant to various domains, including geothermal facilities, hydrocarbon reservoirs, filtration processes, and packed-bed heat exchangers, where inertial forces play a crucial role in flow behavior.
The study explores the influence of several dimensionless parameters on the flow variables, including the velocity ratio, porosity parameter, Forchheimer inertia coefficient, thermal radiation parameter, thermal stratification parameter, and Brinkman number. The range of variables considered includes velocity ratios from 0.1 to 1.6, porosity parameters from 0.1 to 1.2, and radiation parameters from 0.1 to 0.9. These parameters allow the researchers to isolate and examine the specific effects of each physical mechanism on the overall flow and thermal profiles. The analysis provides insights into how these factors interact to determine the performance of the thermal system.
Results indicate that the velocity ratio parameter significantly affects the radial, axial, and tangential velocities. The tri-hybrid nanofluid exhibits the maximum radial velocity compared to hybrid and mono nanofluids, attributed to the augmented momentum transport by the blended nanomaterials. The tangential velocity also increases with the velocity ratio, as the rotational momentum of the fluid across the rotating disk rises with the stretching intensity.
The porosity parameter and Forchheimer inertia coefficient generally reduce the velocity fields. As the permeability factor increases, the radial velocity gradually decreases due to the additional drag force applied by the porous material. This drag lowers the flow rate and dissipates energy from the motion within the surface. Similarly, the Forchheimer term includes inertia friction in the porous medium, which becomes significant at medium and high speeds. The linear resistance rises with the inertia coefficient, causing the fluid velocity to drop and the radial flow to settle down.
Thermal radiation and thermal stratification have opposing effects on the temperature field. The thermal radiation parameter enhances the temperature distribution near the disk and in the ambient fluid, as more heat is radiated into the fluid, boosting heat transport. Conversely, the thermal stratification parameter reduces the temperature field by lessening heat transfer from the surface. This reduction occurs because thermal stratification lowers the temperature differential between the heated surface and the surrounding fluid, thereby diminishing the thermal boundary layer.
The second-law analysis reveals that entropy generation is significantly influenced by the Brinkman number and porous media reactivity. The Brinkman number, which represents viscous dissipation, has a substantial impact on entropy formation. As the irreversible nature owing to viscous forces becomes more prevalent, the Bejan number falls. The Bejan number, which is the ratio of entropy generation due to heat transfer to total entropy generation, declines as the Brinkman number and porosity parameter increase. This indicates that fluid-friction irreversibility plays a larger role than thermal transfer irreversibility in the system.
The tri-hybrid nanofluid demonstrates superior thermal features compared to regular nano and hybrid nanofluids. The GO-Cu-TiO2/engine oil arrangement shows enhanced heat transfer properties and optimized entropy production when contrasted with mono and mixed nanofluid counterparts. Contour visualizations of the speed field confirm that the ternary hybrid nanofluid has the largest area of elevated velocity curves, followed by the hybrid nanofluid. The more homogeneous flow patterns and higher velocity gradients near the spinning surface demonstrate the greater transport capabilities of the tri-hybrid mixture.
The study highlights the potential applications of these findings in coolant features, spinning machines, and electric vehicles. The GO-Cu-TiO2 tri-hybrid arrangement is identified as a desirable choice for heat management tasks in modern rotary machinery and energy technology due to its superior thermal performance. The results provide a strong foundation for the design of more energy-efficient thermal systems. By balancing the velocity ratio, porosity, thermal radiation, and viscous dissipation characteristics, engineers can increase heat transfer while minimizing thermal losses.
The convergence of the semi-analytical solution was verified through h-curves and residual error analysis. The permitted convergence intervals for the auxiliary parameters were identified, ensuring the stability of the series solutions. The residual error decreased with higher orders of approximation, confirming the accuracy of the method. The validation against previous results by Nisar et al. and Hussain further supports the reliability of the mathematical formulation and numerical approaches used in the study.
This research contributes to the understanding of complex fluid dynamics in porous media with multiple nanoparticle types. The integration of thermal radiation, stratification, and entropy analysis offers a comprehensive tool for evaluating thermal system performance. The findings are applicable to various engineering fields where heat transfer and fluid flow are critical. The study underscores the importance of considering inertial effects and thermal non-uniformity in the design of advanced cooling systems.
The mathematical framework relies on specific thermophysical relationships to define the behavior of the tri-hybrid nanofluid. The study defines the viscosity, density, and thermal conductivity of the mixture using equations that account for the volume fractions of the three distinct nanoparticle types. These relationships allow the researchers to quantify how the addition of graphene oxide, copper, and titanium dioxide alters the base properties of the engine oil. The model also incorporates the Prandtl number, which characterizes the ratio of momentum diffusivity to thermal diffusivity, and the radiation parameter, which scales the effect of thermal radiation relative to conduction. By establishing these dimensionless groups, the authors create a standardized basis for comparing the influence of different physical mechanisms on the flow field.
The Homotopy Analysis Method is selected for its ability to handle highly nonlinear equations without requiring small physical parameters. Unlike perturbation approaches, this semi-analytical technique introduces an auxiliary convergence-control parameter that allows for explicit control over the solution series. The researchers determine the optimal values for these parameters by minimizing the average residual error, ensuring that the approximate solution converges to the true physical behavior. The convergence study identifies specific intervals for the auxiliary parameters where the solution curves remain stable and nearly straight, indicating a reliable range for the approximation. This process confirms that the derived series solutions are not only mathematically consistent but also physically meaningful within the defined parameter space.
Validation of the current results is performed by comparing the computed values with previously published data from Nisar et al. and Hussain. The comparison focuses on specific dimensionless parameters, such as the porosity factor and the radiation parameter, under conditions where other variables are set to zero. The numerical data from the current study shows excellent agreement with these earlier findings, which supports the accuracy of the proposed mathematical formulation. This validation step is crucial for establishing the reliability of the new model, as it demonstrates that the semi-analytical approach can reproduce known results for simplified cases before being applied to the more complex tri-hybrid nanofluid scenario.
The study acknowledges that the current model assumes a steady, incompressible, and axisymmetric flow, which simplifies the governing equations but may not capture all transient behaviors. The authors note that the future direction of this research includes examining the transient flow of tri-hybrid nanofluid between two disks, suggesting that the current steady-state analysis is a foundational step rather than a complete description of all possible configurations.
Subject of Research: Semi-analytical analysis of radiative Darcy-Forchheimer flow of tri-hybrid nanofluid over a spinning disk
Article Title: Convergence and residual analysis of radiative Darcy–Forchheimer Homann flow of tri-hybrid nanofluid over a spinning disk with thermal stratification
Article References: Zafar, S. S., Al-Tamimi, S., Ali, A. Y., Saeed, A. M., Ali, F., & Ahmed, M. F. (2026). Convergence and residual analysis of radiative Darcy–Forchheimer Homann flow of tri-hybrid nanofluid over a spinning disk with thermal stratification. Discover Nano, 21(1), Article 573. https://doi.org/10.1186/s11671-026-05029-z
Image Credits: AI Generated
DOI: 10.1186/s11671-026-05029-z
Keywords: tri-hybrid nanofluid, Darcy-Forchheimer flow, spinning disk, thermal stratification, entropy generation, OHAM, heat transfer, porous media, thermal radiation, Discover Nano, Convergence, residual
News Source: Denise Maddox. (November 10, 2026). Tri-hybrid nanofluid flow over spinning disk analyzed. Scienmag.
